Uniformly Convex Functions on Banach Spaces

Given a Banach space (X,ǁ · ǁ), we study the connection between uniformly convex functions f: X → ℝ bounded above by $\|\cdot \|^{p}$ and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f: X → ℝ bounded above by...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2009-03, Vol.137 (3), p.1081-1091
Main Authors: Borwein, J., Guirao, A. J., Hájek, P., Vanderwerff, J.
Format: Article
Language:English
Subjects:
Banach space
Convexity
Exact sciences and technology
Functional analysis
General mathematics
General, history and biography
Mathematical analysis
Mathematical functions
Mathematics
Sciences and techniques of general use
Unit ball
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Summary:Given a Banach space (X,ǁ · ǁ), we study the connection between uniformly convex functions f: X → ℝ bounded above by $\|\cdot \|^{p}$ and the existence of norms on X with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function f: X → ℝ bounded above by ǁ · ǁ² if and only if X admits an equivalent norm with modulus of convexity of power type 2.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09630-5